metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊2D32, D16⋊1D7, C28.5D8, C56.9D4, D112⋊3C2, C14.8D16, C16.4D14, C112.2C22, C7⋊C32⋊1C2, (C7×D16)⋊1C2, C4.1(D4⋊D7), C8.9(C7⋊D4), C2.4(C7⋊D16), SmallGroup(448,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊D32
G = < a,b,c | a7=b32=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C7⋊D32
►On 224 points
Generators in S224
(1 184 79 144 110 218 35)(2 36 219 111 145 80 185)(3 186 81 146 112 220 37)(4 38 221 113 147 82 187)(5 188 83 148 114 222 39)(6 40 223 115 149 84 189)(7 190 85 150 116 224 41)(8 42 193 117 151 86 191)(9 192 87 152 118 194 43)(10 44 195 119 153 88 161)(11 162 89 154 120 196 45)(12 46 197 121 155 90 163)(13 164 91 156 122 198 47)(14 48 199 123 157 92 165)(15 166 93 158 124 200 49)(16 50 201 125 159 94 167)(17 168 95 160 126 202 51)(18 52 203 127 129 96 169)(19 170 65 130 128 204 53)(20 54 205 97 131 66 171)(21 172 67 132 98 206 55)(22 56 207 99 133 68 173)(23 174 69 134 100 208 57)(24 58 209 101 135 70 175)(25 176 71 136 102 210 59)(26 60 211 103 137 72 177)(27 178 73 138 104 212 61)(28 62 213 105 139 74 179)(29 180 75 140 106 214 63)(30 64 215 107 141 76 181)(31 182 77 142 108 216 33)(32 34 217 109 143 78 183)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 186)(34 185)(35 184)(36 183)(37 182)(38 181)(39 180)(40 179)(41 178)(42 177)(43 176)(44 175)(45 174)(46 173)(47 172)(48 171)(49 170)(50 169)(51 168)(52 167)(53 166)(54 165)(55 164)(56 163)(57 162)(58 161)(59 192)(60 191)(61 190)(62 189)(63 188)(64 187)(65 200)(66 199)(67 198)(68 197)(69 196)(70 195)(71 194)(72 193)(73 224)(74 223)(75 222)(76 221)(77 220)(78 219)(79 218)(80 217)(81 216)(82 215)(83 214)(84 213)(85 212)(86 211)(87 210)(88 209)(89 208)(90 207)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 157)(98 156)(99 155)(100 154)(101 153)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 160)(127 159)(128 158)
G:=sub<Sym(224)| (1,184,79,144,110,218,35)(2,36,219,111,145,80,185)(3,186,81,146,112,220,37)(4,38,221,113,147,82,187)(5,188,83,148,114,222,39)(6,40,223,115,149,84,189)(7,190,85,150,116,224,41)(8,42,193,117,151,86,191)(9,192,87,152,118,194,43)(10,44,195,119,153,88,161)(11,162,89,154,120,196,45)(12,46,197,121,155,90,163)(13,164,91,156,122,198,47)(14,48,199,123,157,92,165)(15,166,93,158,124,200,49)(16,50,201,125,159,94,167)(17,168,95,160,126,202,51)(18,52,203,127,129,96,169)(19,170,65,130,128,204,53)(20,54,205,97,131,66,171)(21,172,67,132,98,206,55)(22,56,207,99,133,68,173)(23,174,69,134,100,208,57)(24,58,209,101,135,70,175)(25,176,71,136,102,210,59)(26,60,211,103,137,72,177)(27,178,73,138,104,212,61)(28,62,213,105,139,74,179)(29,180,75,140,106,214,63)(30,64,215,107,141,76,181)(31,182,77,142,108,216,33)(32,34,217,109,143,78,183), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,186)(34,185)(35,184)(36,183)(37,182)(38,181)(39,180)(40,179)(41,178)(42,177)(43,176)(44,175)(45,174)(46,173)(47,172)(48,171)(49,170)(50,169)(51,168)(52,167)(53,166)(54,165)(55,164)(56,163)(57,162)(58,161)(59,192)(60,191)(61,190)(62,189)(63,188)(64,187)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,224)(74,223)(75,222)(76,221)(77,220)(78,219)(79,218)(80,217)(81,216)(82,215)(83,214)(84,213)(85,212)(86,211)(87,210)(88,209)(89,208)(90,207)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,160)(127,159)(128,158)>;
G:=Group( (1,184,79,144,110,218,35)(2,36,219,111,145,80,185)(3,186,81,146,112,220,37)(4,38,221,113,147,82,187)(5,188,83,148,114,222,39)(6,40,223,115,149,84,189)(7,190,85,150,116,224,41)(8,42,193,117,151,86,191)(9,192,87,152,118,194,43)(10,44,195,119,153,88,161)(11,162,89,154,120,196,45)(12,46,197,121,155,90,163)(13,164,91,156,122,198,47)(14,48,199,123,157,92,165)(15,166,93,158,124,200,49)(16,50,201,125,159,94,167)(17,168,95,160,126,202,51)(18,52,203,127,129,96,169)(19,170,65,130,128,204,53)(20,54,205,97,131,66,171)(21,172,67,132,98,206,55)(22,56,207,99,133,68,173)(23,174,69,134,100,208,57)(24,58,209,101,135,70,175)(25,176,71,136,102,210,59)(26,60,211,103,137,72,177)(27,178,73,138,104,212,61)(28,62,213,105,139,74,179)(29,180,75,140,106,214,63)(30,64,215,107,141,76,181)(31,182,77,142,108,216,33)(32,34,217,109,143,78,183), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,186)(34,185)(35,184)(36,183)(37,182)(38,181)(39,180)(40,179)(41,178)(42,177)(43,176)(44,175)(45,174)(46,173)(47,172)(48,171)(49,170)(50,169)(51,168)(52,167)(53,166)(54,165)(55,164)(56,163)(57,162)(58,161)(59,192)(60,191)(61,190)(62,189)(63,188)(64,187)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,224)(74,223)(75,222)(76,221)(77,220)(78,219)(79,218)(80,217)(81,216)(82,215)(83,214)(84,213)(85,212)(86,211)(87,210)(88,209)(89,208)(90,207)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,160)(127,159)(128,158) );
G=PermutationGroup([[(1,184,79,144,110,218,35),(2,36,219,111,145,80,185),(3,186,81,146,112,220,37),(4,38,221,113,147,82,187),(5,188,83,148,114,222,39),(6,40,223,115,149,84,189),(7,190,85,150,116,224,41),(8,42,193,117,151,86,191),(9,192,87,152,118,194,43),(10,44,195,119,153,88,161),(11,162,89,154,120,196,45),(12,46,197,121,155,90,163),(13,164,91,156,122,198,47),(14,48,199,123,157,92,165),(15,166,93,158,124,200,49),(16,50,201,125,159,94,167),(17,168,95,160,126,202,51),(18,52,203,127,129,96,169),(19,170,65,130,128,204,53),(20,54,205,97,131,66,171),(21,172,67,132,98,206,55),(22,56,207,99,133,68,173),(23,174,69,134,100,208,57),(24,58,209,101,135,70,175),(25,176,71,136,102,210,59),(26,60,211,103,137,72,177),(27,178,73,138,104,212,61),(28,62,213,105,139,74,179),(29,180,75,140,106,214,63),(30,64,215,107,141,76,181),(31,182,77,142,108,216,33),(32,34,217,109,143,78,183)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,186),(34,185),(35,184),(36,183),(37,182),(38,181),(39,180),(40,179),(41,178),(42,177),(43,176),(44,175),(45,174),(46,173),(47,172),(48,171),(49,170),(50,169),(51,168),(52,167),(53,166),(54,165),(55,164),(56,163),(57,162),(58,161),(59,192),(60,191),(61,190),(62,189),(63,188),(64,187),(65,200),(66,199),(67,198),(68,197),(69,196),(70,195),(71,194),(72,193),(73,224),(74,223),(75,222),(76,221),(77,220),(78,219),(79,218),(80,217),(81,216),(82,215),(83,214),(84,213),(85,212),(86,211),(87,210),(88,209),(89,208),(90,207),(91,206),(92,205),(93,204),(94,203),(95,202),(96,201),(97,157),(98,156),(99,155),(100,154),(101,153),(102,152),(103,151),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,160),(127,159),(128,158)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14I | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 32A | ··· | 32H | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 16 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 4 | ··· | 4 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | C7⋊D4 | D32 | D4⋊D7 | C7⋊D16 | C7⋊D32 |
| kernel | C7⋊D32 | C7⋊C32 | D112 | C7×D16 | C56 | D16 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 3 | 6 | 12 |
Matrix representation of C7⋊D32 ►in GL4(𝔽449) generated by
| 0 | 1 | 0 | 0 |
| 448 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 43 | 448 | 0 | 0 |
| 0 | 0 | 302 | 198 |
| 0 | 0 | 379 | 247 |
| 1 | 0 | 0 | 0 |
| 43 | 448 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 324 | 448 |
G:=sub<GL(4,GF(449))| [0,448,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[1,43,0,0,0,448,0,0,0,0,302,379,0,0,198,247],[1,43,0,0,0,448,0,0,0,0,1,324,0,0,0,448] >;
C7⋊D32 in GAP, Magma, Sage, TeX
C_7\rtimes D_{32} % in TeX
G:=Group("C7:D32"); // GroupNames label
G:=SmallGroup(448,76);
// by ID
G=gap.SmallGroup(448,76);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,254,135,142,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^7=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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